Famous Norwegian mathematician Niels Henrik Abel advised that one should "learn from the masters, not from the pupils". When the subject is algebraic numbers and algebraic functions, there is no greater master than Emil Artin. In this classic text, originated from the notes of the course given at Princeton University in 1950-1951 and first published in 1967, one has a beautiful introduction to the subject accompanied by Artin's unique insights and perspectives. The exposition starts with the general theory of valuation fields in Part I, proceeds to the local class field theory in Part II, and then to the theory of function fields in one variable (including the Riemann-Roch theorem and its applications) in Part III.

Prerequisites for reading the book are a standard first-year graduate course in algebra (including some Galois theory) and elementary notions of point set topology. With many examples, this book can be used by graduate students and all mathematicians learning number theory and related areas of algebraic geometry of curves.

Readership

Graduate students and research mathematicians interested in number theory and algebraic geometry.

Reviews

"The exposition is (as usual with Artin) quite elegant, and the parallel treatment of number fields and function can be illuminating as well as efficient ... a master of the subject ... It is a true classic in the field."

-- MAA Reviews

"Now, after another forty years, and being out of print for the last decades, Artin's classic of timeless beauty has been made available again for new generations of students, teachers, researchers, mathematics historians, and bibliophiles, very much to the benefit of the mathematical community as a whole."

-- Zentralblatt MATH